gganimate with lines.

General
2

I can't be sure this will work for you with a reprex, but I think you need to use transition_reveal instead of transition_time for geom_path/geom_line.

So you would do:

p <- ggplot(exim1, aes(x = Year, y = value, color = Country)) +
  geom_path() +
  geom_point() +
  facet_grid(Flow ~ Country) +
  theme(legend.position = 'none') +
  labs(title = 'GDP per Capita, Year: {frame_time}') +
  transition_reveal(id = Year, along = Year) +
  ease_aes('linear')

animate(p, 100, 10)

You have to pass the Year column name to both the id and along arguments of the function.